Homogeneous cosmological models

The most drastic symmetry assumption which still allows some dynamics is
that of spatially homogeneity. In that case the partial differential
equations reduce to ordinary differential equations since all
variables depend on time alone. In general relativity the gravitational
field has interesting and complicated dynamics even in the absence of
matter ("vacuum case"). The homogeneous vacuum Einstein equations lead to
a system of four ordinary differential equations (the Wainwright-Hsu
system). When the simplest type of matter, a perfect fluid, is included
the number of equations is increased to five. The dynamics of general
solutions of this system is still far from understood, despite major
advances by Ringström in the late 1990's. I have a project with the
group of
Bernold Fiedler at the Free University in Berlin to bring
sophisticated techniques from the theory of dynamical systems to bear on this
subject. This is project B7 of the
special research area SFB
647, Space-Time-Matter. It has been hypothesized that the dynamics is
controlled by an object called the Mixmaster attractor. The aim of this
project is to find out in what sense and to what extent this is true.
It seems a priori difficult to obtain information about the dynamics of
general solutions of a specific five-dimensional dynamical system. That
it is possible for the WH system is due to some very special properties.
It has some lower dimensional invariant submanifolds and there are
Liapunov functions which drive the solution towards these submanifolds
during the evolution. Thus much of the evolution of general solutions
can be encoded in terms of the dynamics on the submanifolds. It is
nevertheless the case that these lower dimensional dynamical systems,
which themselves have relatively simple behaviour, combine to produce very
subtle effects. This is why it has taken the research community so long to
make progress on this problem.
An interesting type of matter source for the gravitational field, both for
physics and for mathematics, is collisionless matter described by kinetic
theory. For this matter model the basic unknown depends on time, position
and velocity. The spatial dependence is eliminated by spatial homogeneity
but the velocity dependence remains. Thus in this case a partial
differential equation is retained even in the homogeneous case. There are
special cases where the problem can be reduced to a system of ordinary
differential equations and their dynamics have been analysed by Paul Tod,
Claes Uggla and myself, cf.
gr-qc/9811051 ,
gr-qc/0005116 ,
and
gr-qc/0112040 .
To go beyond this restricted situation is the goal of the PhD project of
Ernesto Nungesser.